Monad (category theory) If and are a pair of adjoint functors, with left adjoint to , then the composition is a monad. are inverse functors, the corresponding monad is the identity functor. , is discussed under the dual theory of comonads. The monad axioms can be seen at work in a simple example: let we can take the free group functor. This means that the monad takes a set and returns the underlying set of the free group . by including any set in in the natural way, as strings of length 1. can be made out of a natural concatenation or 'flattening' of 'strings of strings'. They will satisfy some axioms about identity and associativity that result from the adjunction properties. Those axioms are formally similar to the monoid axioms. If we specialize to categories arising from partially ordered sets (with a single morphism from to iff ), then the formalism becomes much simpler: adjoint pairs are Galois connections and monads are closure operators. Every monad arises from some adjunction, in fact typically from many adjunctions.
Kategorientheorie Bedeutung[Bearbeiten] Aufgrund ihres hohen Grades an Abstraktion wird die Kategorientheorie gelegentlich – selbst von den Mathematikern, die sie entwickelten – als allgemeiner Unsinn bezeichnet.[1][2] Definitionen[Bearbeiten] Kategorie[Bearbeiten] Eine Kategorie besteht aus folgendem: die im offensichtlichen Sinne assoziativ sind: sofern und (Gelegentlich wird das weggelassen und als angeschrieben.) einem Identitätsmorphismus zu jedem Objekt , der neutrales Element für die Verknüpfung mit Morphismen mit Quelle oder Ziel ist, d. h. es gilt , falls ist, und , falls . Die Klasse aller Morphismen wird auch mit oder bezeichnet (von englisch arrow, französisch flèche, deutsch Pfeil). Unterkategorie[Bearbeiten] Eine Unterkategorie einer Kategorie ist eine Kategorie , so dass eine Teilklasse von ist und für je zwei Objekte in die Morphismenmenge Teilmenge von ist. gleich denen von , ist eine volle Unterkategorie. Duale Kategorie[Bearbeiten] Die duale Kategorie zu einer Kategorie ist die Kategorie mit . ist gleich mit nach
Background and genesis of topos theory This page gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the other hand context can be given. This is partly in terms of historical development, but also to some extent an explanation of differing attitudes to category theory. In the school of Grothendieck[edit] During the latter part of the 1950s, the foundations of algebraic geometry were being rewritten; and it is here that the origins of the topos concept are to be found. With the benefit of hindsight, it can be said that algebraic geometry had been wrestling with two problems for a long time. Such a definition of a topos was eventually given five years later, around 1962, by Grothendieck and Verdier (see Verdier's Bourbaki seminar Analysis Situs). From pure category theory to categorical logic[edit] Position of topos theory[edit] Summary[edit]
Groupoid In mathematics , especially in category theory and homotopy theory , a groupoid (less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: Special cases include: Setoids , that is: sets that come with an equivalence relation ; G-sets , sets equipped with an action of a group G . Groupoids are often used to reason about geometrical objects such as manifolds . Definitions [ edit ] Algebraic [ edit ] A groupoid is a set G with a unary operation and a partial function Here * is not a binary operation because it is not necessarily defined for all possible pairs of G -elements. and −1 have the following axiomatic properties. Associativity : If a * b and b * c are defined, then ( a * b ) * c and a * ( b * c ) are defined and equal. In short: ( a * b ) * c = a * ( b * c ); ∀a ∃ a −1 * a ∃ a * a −1 ; a * b * b −1 = a and a −1 * a * b = b . From these axioms, two easy and convenient properties follow: Category theoretic [ edit ]
Scientific method For observational studies see observational study An 18th-century depiction of early experimentation in the field of chemistry The scientific method is a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge.[1] To be termed scientific, a method of inquiry must be based on empirical and measurable evidence subject to specific principles of reasoning.[2] The Oxford English Dictionary defines the scientific method as: "a method or procedure that has characterized natural science since the 17th century, consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses."[3] The chief characteristic which distinguishes the scientific method from other methods of acquiring knowledge is that scientists seek to let reality speak for itself,[discuss] supporting a theory when a theory's predictions are confirmed and challenging a theory when its predictions prove false.
Thue (programming language) A Thue program starts with a rulebase, which is a series of substitution rules, each of this form: lhs ::= rhs The rulebase terminates with a lone production symbol on a line: The initial state is a series of symbols which follow the rulebase. Thue consumes the initial symbols and substitutes the result of the rules for each of the initial state's symbols. Thue terminates when lhs cannot be found in a resultant state. ::= is pronounced can be.lhs is "left hand side".rhs is "right hand side"."::=" can never be the lhs.":::" is an input stream."~" is the output stream.Semi-Thue systems are isomorphic to unrestricted grammars. When invoked with 'd' (debug), print the state. Here's the traditional "Hello World!" a::=~Hello World! The following Thue program performs an increment of a binary number entered as the initial state surrounded by "_" characters, in this case the number 1111111111: b::=~0 b::=~1 ac::=abc ::= abc
Monad (functional programming) Formally, a monad consists of a type constructor M and two operations, bind and return (where return is often also called unit). The operations must fulfill several properties to allow the correct composition of monadic functions (i.e. functions that use values from the monad as their arguments or return value). The return operation takes a value from a plain type and puts it into a monadic container using the constructor, creating a monadic value. The bind operation performs the reverse process, extracting the original value from the container and passing it to the associated next function in the pipeline, possibly with additional checks and transformations. Because a monad can insert additional operations around a program's domain logic, monads can be considered a sort of aspect-oriented programming.[7] The domain logic can be defined by the application programmer in the pipeline, while required aside bookkeeping operations can be handled by a pre-defined monad built in advance.
Haskell (programming language) Functional programming language Haskell is used in academia and industry.[29][30][31] As of May 2021[update], Haskell was the 28th most popular programming language by Google searches for tutorials,[32] and made up less than 1% of active users on the GitHub source code repository.[33] Haskell was developed by a committee, attempting to bring together off the shelf solutions where possible. Type classes, which enable type-safe operator overloading, were first proposed by Philip Wadler and Stephen Blott to address the ad-hoc handling of equality types and arithmetic overloading in languages at the time.[35] Other notable changes in early versions were the approach to the 'seq' function, which creates a data dependency between values, and is used in lazy languages to avoid excessive memory consumption; with it moving from a type class to a standard function to make refactoring more practical. The first version of Haskell ("Haskell 1.0") was defined in 1990. A "Hello, World!" [edit] Reports